Focus on Fluids
Waving plants and turbulent eddies
- J. J. FINNIGAN
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- Published online by Cambridge University Press:
- 19 May 2010, pp. 1-4
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New large-eddy simulations of flow over a flexible plant canopy by Dupont et al. (J. Fluid Mech., 2010, this issue, vol. 652, pp. 5–44) have produced apparently paradoxical results. Work over the last three decades had suggested that turbulent eddies could ‘lock onto’ to the waving frequency of uniform cereal canopies. Their new simulations contradict this view, although a resolution may lie in the essentially three-dimensional nature of the instability process that generates the dominant eddies above plant canopies.
Papers
Modelling waving crops using large-eddy simulation: comparison with experiments and a linear stability analysis
- S. DUPONT, F. GOSSELIN, C. PY, E. DE LANGRE, P. HEMON, Y. BRUNET
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- 15 April 2010, pp. 5-44
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In order to investigate the possibility of modelling plant motion at the landscape scale, an equation for crop plant motion, forced by an instantaneous velocity field, is introduced in a large-eddy simulation (LES) airflow model, previously validated over homogeneous and heterogeneous canopies. The canopy is simply represented as a poroelastic continuous medium, which is similar in its discrete form to an infinite row of identical oscillating stems. Only one linear mode of plant vibration is considered. Two-way coupling between plant motion and the wind flow is insured through the drag force term. The coupled model is validated on the basis of a comparison with measured movements of an alfalfa crop canopy. It is also compared with the outputs of a linear stability analysis. The model is shown to reproduce the well-known phenomenon of ‘honami’ which is typical of wave-like crop motions on windy days. The wavelength of the main coherent waving patches, extracted using a bi-orthogonal decomposition (BOD) of the crop velocity fields, is in agreement with that deduced from video recordings. The main spatial and temporal characteristics of these waving patches exhibit the same variation with mean wind velocity as that observed with the measurements. However they differ from the coherent eddy structures of the wind flow at canopy top, so that coherent waving patches cannot be seen as direct signatures of coherent eddy structures. Finally, it is shown that the impact of crop motion on the wind dynamics is negligible for current wind speed values. No lock-in mechanism of coherent eddy structures on plant motion is observed, in contradiction with the linear stability analysis. This discrepancy may be attributed to the presence of a nonlinear saturation mechanism in LES.
Three-dimensional direct numerical simulation of a turbulent lifted hydrogen jet flame in heated coflow: a chemical explosive mode analysis
- T. F. LU, C. S. YOO, J. H. CHEN, C. K. LAW
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- 19 May 2010, pp. 45-64
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A chemical explosive mode analysis (CEMA) was developed as a new diagnostic to identify flame and ignition structure in complex flows. CEMA was then used to analyse the near-field structure of the stabilization region of a turbulent lifted hydrogen–air slot jet flame in a heated air coflow computed with three-dimensional direct numerical simulation. The simulation was performed with a detailed hydrogen–air mechanism and mixture-averaged transport properties at a jet Reynolds number of 11000 with over 900 million grid points. Explosive chemical modes and their characteristic time scales, as well as the species involved, were identified from the Jacobian matrix of the chemical source terms for species and temperature. An explosion index was defined for explosive modes, indicating the contribution of species and temperature in the explosion process. Radical and thermal runaway can consequently be distinguished. CEMA of the lifted flame shows the existence of two premixed flame fronts, which are difficult to detect with conventional methods. The upstream fork preceding the two flame fronts thereby identifies the stabilization point. A Damköhler number was defined based on the time scale of the chemical explosive mode and the local instantaneous scalar dissipation rate to highlight the role of auto-ignition in affecting the stabilization points in the lifted jet flame.
Scaling of global properties of turbulence and skin friction in pipe and channel flows
- VICTOR YAKHOT, SEAN C. C. BAILEY, ALEXANDER J. SMITS
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- 19 May 2010, pp. 65-73
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Experimental data on the Reynolds number dependence of the area-averaged turbulent kinetic energy K and dissipation rate ℰ are presented. It is shown that while in the interval ReD > 105 the total kinetic energy scales with friction velocity (K/u*2 = const), a new scaling law K/〈U〉2 ∝ K/(u*2ReDθ) = const (θ ≈ 1/4) has been discovered in the interval ReD < 105. It is argued that this transition is responsible for the well-known change in the scaling behaviour of the friction factor observed in pipe and channels flows at ReD ≈ 105.
Large-eddy simulation of a buoyant plume in uniform and stably stratified environments
- B. J. DEVENISH, G. G. ROONEY, D. J. THOMSON
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- 09 April 2010, pp. 75-103
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We consider large-eddy simulation (LES) of buoyant plumes in uniform and stably stratified environments. We show that in the former case the results agree well with the simple plume model of Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, p. 1). In particular, we calculate an entrainment constant which is consistent with laboratory and field measurements and find no significant difference between the radial spreading rates of vertical velocity and buoyancy. In a stably stratified environment, the LES plume shows better agreement with Morton et al. (1956) below the level at which the buoyancy first vanishes than above this level. Above the level of neutral buoyancy, the LES plume is characterized by an ascending core of negative buoyancy surrounded by a descending annulus of positive buoyancy. We compare the LES data with the model of Bloomfield & Kerr (J. Fluid Mech., vol. 424, 2000, p. 197), which explicitly accounts for these coherent motions. The model exhibits many qualitative aspects of the LES plume and quantitative agreement can be improved by adjusting the downward volume flux relative to the upward volume flux in a manner consistent with the LES plume. This simple adjustment, along with revised values of the entrainment constants, represents the combined effects of an overturning region at the top of the plume (where a fluid element reverses direction), ‘plume-top’ entrainment (whereby the plume entrains ambient fluid above the plume) as well as lateral entrainment and detrainment processes (both external and internal) occurring above the top of the model plume.
Large particle segregation, transport and accumulation in granular free-surface flows
- J. M. N. T. GRAY, B. P. KOKELAAR
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- 19 May 2010, pp. 105-137
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Particle size segregation can have a significant feedback on the motion of many hazardous geophysical mass flows such as debris flows, dense pyroclastic flows and snow avalanches. This paper develops a new depth-averaged theory for segregation that can easily be incorporated into the existing depth-averaged structure of typical models of geophysical mass flows. The theory is derived by depth-averaging the segregation-remixing equation for a bi-disperse mixture of large and small particles and assuming that (i) the avalanche is always inversely graded and (ii) there is a linear downslope velocity profile through the avalanche depth. Remarkably, the resulting ‘large particle transport equation’ is very closely related to the segregation equation from which it is derived. Large particles are preferentially transported towards the avalanche front and then accumulate there. This is important, because when this is combined with mobility feedback effects, the larger less mobile particles at the front can be continuously shouldered aside to spontaneously form lateral levees that channelize the flow and enhance run-out. The theory provides a general framework that will enable segregation-mobility feedback effects to be studied in detail for the first time. While the large particle transport equation has a very simple representation of the particle size distribution, it does a surprisingly good job of capturing solutions to the full theory once the grains have segregated into inversely graded layers. In particular, we show that provided the inversely graded interface does not break it has precisely the same solution as the full theory. When the interface does break, a concentration shock forms instead of a breaking size segregation wave, but the net transport of large particles towards the flow front is exactly the same. The theory can also model more complex effects in small-scale stratification experiments, where particles may either be brought to rest by basal deposition or by the upslope propagation of a granular bore. In the former case the resulting deposit is normally graded, while in the latter case it is inversely graded. These completely opposite gradings in the deposit arise from a parent flow that is inversely graded, which raises many questions about how to interpret geological deposits.
Self-conditioned fields for large-eddy simulations of turbulent flows
- STEPHEN B. POPE
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- 13 April 2010, pp. 139-169
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An alternative foundation is developed for the large-eddy simulation (LES) of turbulent flows. It is based on ‘self-conditioned fields’, for example, the mean velocity field conditional on a discrete representation of the filtered velocity field. It is shown that the self-conditioned velocity field minimizes the residual kinetic energy, and that, with the ideal model, the method yields the correct one-time behaviour as determined by the Navier–Stokes equations. The approach is extended to the self-conditioned probability density function (PDF) of compositions. Compared to LES formulations based on the filtered velocity and the filtered density function, the self-conditioned field approach has several advantages: for laminar flow, and in the direct-numerical-simulation limit, the residual fluctuations are zero or exponentially small; full account is taken of the probability distribution of turbulent fields; there are no commutation issues; and there are no issues with filtering at walls, where the self-conditioned velocity is zero. The exact evolution equations for the self-conditioned velocity and composition PDF are derived. Basic models are presented, and the development of improved models is discussed.
Viscous symmetric stability of circular flows
- R. C. KLOOSTERZIEL
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- 19 May 2010, pp. 171-193
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The linear stability properties of viscous circular flows in a rotating environment are studied with respect to symmetric perturbations. Through the use of an effective energy or Lyapunov functional, we derive sufficient conditions for Lyapunov stability with respect to such perturbations. For circular flows with swirl velocity V(r) we find that Lyapunov stability is determined by the properties of the function ℱ(r) = (2V/r + f)/Q (with f the Coriolis parameter, r the radius and Q the absolute vorticity) instead of the customary Rayleigh discriminant Φ(r) = (2V/r + f)Q. The conditions for stability are valid for flows with non-zero Q everywhere. Further, the flows are presumed stationary, incompressible and velocity perturbations are required to vanish at rigid boundaries. For Lyapunov stable flows an upper bound for the increase of the total perturbation energy due to transient non-modal growth is derived which is valid for any Reynolds number. The theory is applied to Couette flow and the Lamb–Oseen vortex.
The role of diffusion on the interface thickness in a ventilated filling box
- N. B. KAYE, M. R. FLYNN, M. J. COOK, Y. JI
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- 09 April 2010, pp. 195-205
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We examine the role of diffusivity, whether molecular or turbulent, on the steady-state stratification in a ventilated filling box. The buoyancy-driven displacement ventilation model of Linden et al. (J. Fluid Mech., vol. 212, 1990, p. 309) predicts the formation of a two-layer stratification when a single plume is introduced into an enclosure with vents at the top and bottom. The model assumes that diffusion plays no role in the development of the ambient buoyancy stratification: diffusion is a slow process and the entrainment of ambient fluid into the plume from the diffuse interface will act to thin the interface resulting in a near discontinuity of density between the upper and lower layers. This prediction has been corroborated by small-scale salt bath experiments; however, full-scale measurements in ventilated rooms and complementary numerical simulations suggest an interface that is not sharp but rather smeared out over a finite thickness. For a given plume buoyancy flux, as the cross-sectional area of the enclosure increases the volume of fluid that must be entrained by the plume to maintain a sharp interface also increases. Therefore the balance between the diffusive thickening of the interface and plume-driven thinning favours a thicker interface. Conversely, the interface thickness decreases with increasing source buoyancy flux, although the dependence is relatively weak. Our analysis presents two models for predicting the interface thickness as a function of the enclosure height, base area, composite vent area, plume buoyancy flux and buoyancy diffusivity. Model results are compared with interface thickness measurements based on previously reported data. Positive qualitative and quantitative agreement is observed.
Mechanics of liquid–liquid interfaces and mixing enhancement in microscale flows
- STÉPHANE VERGUET, CHUANHUA DUAN, ALBERT LIAU, VEYSEL BERK, JAMIE H. D. CATE, ARUN MAJUMDAR, ANDREW J. SZERI
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- 19 May 2010, pp. 207-240
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Experimental work on mixing in microfluidic devices has been of growing importance in recent years. Interest in probing reaction kinetics faster than the minute or hour time scale has intensified research in designing microchannel devices that would allow the reactants to be mixed on a time scale faster than that of the reaction. Particular attention has been paid to the design of microchannels in order to enhance the advection phenomena in these devices. Ultimately, in vitro studies of biological reactions can now be performed in conditions that reflect their native intracellular environments. Liau et al. (Anal. Chem., vol. 77, 2005, p. 7618) have demonstrated a droplet-based microfluidic mixer that induces improved chaotic mixing of crowded solutions in milliseconds due to protrusions (‘bumps’) on the microchannel walls. Liau et al. (2005) have shown it to be possible to mix rapidly plugs of highly concentrated protein solutions such as bovine hemoglobin and bovine serum albumin. The present work concerns an analysis of the underlying mechanisms of shear stress transfer at liquid–liquid interfaces and associated enhanced mixing arising from the protrusions along the channel walls. The role of non-Newtonian rheology and surfactants is also considered within the mixing framework developed by Aref, Ottino and Wiggins in several publications. Specifically, we show that proportional thinning of the carrier fluid lubrication layer at the bumps leads to greater advection velocities within the plugs, which enhances mixing. When the fluid within the plugs is Newtonian, mixing will be enhanced by the bumps if they are sufficiently close to one another. Changing either the rheology of the fluid within the plugs (from Newtonian to non-Newtonian) or modifying the mechanics of the carrier fluid-plug interface (by populating it with insoluble surfactants) alters the mixing enhancement.
Lattice Boltzmann simulation of electromechanical resonators in gaseous media
- CARLOS E. COLOSQUI, DEVREZ M. KARABACAK, KAMIL L. EKINCI, VICTOR YAKHOT
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- 30 March 2010, pp. 241-257
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In this work, we employ a kinetic-theory-based approach to predict the hydrodynamic forces on electromechanical resonators operating in gaseous media. Using the Boltzmann–BGK equation, we investigate the influence of the resonator geometry on the fluid resistance in the entire range of non-dimensional frequency variation 0 ≤ τω ≤ ∞; here the fluid relaxation time τ = μ/p is determined by the gas viscosity μ and pressure p at thermodynamic equilibrium, and ω is the (angular) oscillation frequency. Our results here capture two important aspects of recent experimental measurements that covered a broad range of experimental parameters. First, the experimentally observed transition from viscous to viscoelastic flow in simple gases at τω ≈ 1 emerges naturally in the numerical data. Second, the calculated effects of resonator geometry are in agreement with experimental observations.
Fully nonlinear periodic internal waves in a two-fluid system of finite depth
- R. CAMASSA, P.-O. RUSÅS, A. SAXENA, R. TIRON
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- 19 May 2010, pp. 259-298
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Periodic travelling wave solutions for a strongly nonlinear model of long internal wave propagation in a two-fluid system are derived and extensively analysed, with the aim of providing structure to the rich parametric space of existence of such waves for the parent Euler system. The waves propagate at the interface between two homogeneous-density incompressible fluids filling the two-dimensional domain between rigid planar boundaries. The class of waves with a prescribed mean elevation, chosen to coincide with the origin of the vertical (parallel to gravity) axis, and prescribed zero period-average momentum and volume-flux is studied in detail. The constraints are selected because of their physical interpretation in terms of possible processes of wave generation in wave-tanks, and give rise to a quadrature formula which is analysed in parameter space with a combination of numerical and analytical tools. The resulting model solutions are validated against those computed numerically from the parent Euler two-layer system with a boundary element method. The parametric domain of existence of model periodic waves is determined in closed form by curves in the amplitude–speed (A, c) parameter plane corresponding to infinite period limiting cases of fronts (conjugate states) and solitary waves. It is found that the existence domain of Euler solutions is a subset of that of the model. A third closed form relation between c and A indicates where the Euler solutions cease to exist within the model's domain, and this is related to appearance of ‘overhanging’ (multiple valued) wave profiles. The model existence domain is further partitioned in regions where the model is expected to provide accurate approximations to Euler solutions based on analytical estimates from the quadrature. The resulting predictions are found to be in good agreement with the numerical Euler solutions, as exhibited by several wave properties, including kinetic and potential energy, over a broad range of parameter values, extending to the limiting cases of critical depth ratio and extreme density ratios. In particular, when the period is sufficiently long, model solutions show that for a given supercritical speed waves of substantially larger amplitude than the limiting amplitude of solitary waves can exist, and are good approximations of the corresponding Euler solutions. This finding can be relevant for modelling field observations of oceanic internal waves, which often occur in wavetrains with multiple peaks.
On the boundary-layer structure of high-Prandtl-number horizontal convection
- P. G. DANIELS
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- 19 May 2010, pp. 299-331
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This paper describes the boundary-layer structure of the steady flow of an infinite Prandtl number fluid in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated and the upper surface is assumed to be either shear-free or rigid. In the limit of large Rayleigh number (R → ∞), the solution involves a horizontal boundary layer at the upper surface of depth of order R−1/5 where the main variation in the temperature field occurs. For a monotonic temperature distribution at the upper surface, fluid is driven to the colder end of the cavity where it descends within a narrow convection-dominated vertical layer before returning to the horizontal layer. A numerical solution of the horizontal boundary-layer problem is found for the case of a linear temperature distribution at the upper surface. At greater depths, of order R−2/15 for a shear-free surface and order R−9/65 for a rigid upper surface, a descending plume near the cold sidewall, together with a vertically stratified interior flow, allow the temperature to attain an approximately constant value throughout the remainder of the cavity. For a shear-free upper surface, this constant temperature is predicted to be of order R−1/15 higher than the minimum temperature of the upper surface, whereas for a rigid upper surface it is predicted to be of order R−2/65 higher.
DNS of wavepacket evolution in a Blasius boundary layer
- K. S. YEO, X. ZHAO, Z. Y. WANG, K. C. NG
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- 19 May 2010, pp. 333-372
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This paper presents the direct numerical simulation (DNS) of wavepacket evolution and breakdown in a Blasius boundary layer. The study covers the physical, spectral and structural aspects of the whole transition process, whereas previous studies have tended to focus on issues of a more limited scope. The simulations are modelled after the experiments of Cohen, Breuer & Haritonidis (J. Fluid Mech., vol. 225, 1991, p. 575). The disturbance wavepackets are initiated here by a u-velocity and a v-velocity delta pulse. They evolve through a quasi-linear growth stage, a subharmonic stage and a strongly nonlinear stage before breaking down into the nascent turbulent spots. Pulse-initiated wavepackets provide a plausible model for naturally occurring laminar–turbulent transition because they contain disturbances in a broadband of frequencies and wavenumbers, whose sum of interactions determines the spatio-temporal progress of the wavepackets. The early development of the wavepackets accords well with established linear results. The ensuing subharmonic evolution of the wavepackets appears to be underpinned by a critical-layer-based mechanism in which the x-phase speeds of the fundamental two-dimensional and dominant three-dimensional waves with compatible Squire wavenumbers are approximately matched. Spectral data over the bulk of the subharmonic stage demonstrate good consistency with the action of a phase-locked theory recently proposed by Wu, Stewart & Cowley (J. Fluid Mech., vol. 590, 2007, p. 265), strongly suggesting that the latter may be the dominant mechanism in the broadband nonlinear evolution of wavepackets. The dominant two-dimensional and three-dimensional waves are observed to be spontaneously evolving towards triad resonance in the late subharmonic stage. The simulations reproduce many key features in the experiments of Cohen et al. (1991) and Medeiros & Gaster (J. Fluid Mech., vol. 399, 1999b, p. 301). A plausible explanation is also offered for the apparently ‘deterministic’ subharmonic behaviour of wavepackets observed by Medeiros & Gaster. The strongly nonlinear stage is signified by the appearance of low-frequency streamwise-aligned u-velocity structures at twice the spanwise wavenumber of the dominant three-dimensional waves, distortion of the local base flow by the strengthening primary Λ-vortex and rapid expansion of the spanwise wavenumber (β) spectrum. These are in broad agreement with the experimental observations of Breuer, Cohen & Haritonidis (J. Fluid Mech., vol. 340, 1997, p. 395). The breakdown into incipient turbulent spots occurs at locations consistent with the experiments of Cohen et al. (1991). A visualization shows that the evolving wavepackets comprise very thin overlapping vorticity sheets of alternating signs, in stacks of two or three. Strong streamwise stretching of the flow at the centre of the wavepacket in the late subharmonic and strongly nonlinear stages promotes the roll-up and intensification of the vorticity sheets into longitudinal vortices, whose mutual induction precedes the breakdown of the wavepacket. The critical layer of the dominant two-dimensional and oblique wave modes reveals the progressive coalescence of a strong pair of vortices (associated with the Λ-vortex) during the subharmonic stage. Their coalescence culminates in a strong upward burst of velocity that transports lower momentum fluid from below the critical layer into the upper boundary layer to form a high shear layer in the post-subharmonic stage.
A comparative study of self-propelled and towed wakes in a stratified fluid
- KYLE A. BRUCKER, SUTANU SARKAR
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- 12 April 2010, pp. 373-404
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Direct numerical simulations (DNS) of axisymmetric wakes with canonical towed and self-propelled velocity profiles are performed at Re = 50 000 on a grid with approximately 2 billion grid points. The present study focuses on a comparison between towed and self-propelled wakes and on the elucidation of buoyancy effects. The development of the wake is characterized by the evolution of maxima, area integrals and spatial distributions of mean and turbulence statistics. Transport equations for mean and turbulent energies are utilized to help understand the observations. The mean velocity in the self-propelled wake decays more rapidly than the towed case due to higher shear and consequently a faster rate of energy transfer to turbulence. Buoyancy allows a wake to survive longer in a stratified fluid by reducing the 〈u1′u3′〉 correlation responsible for the mean-to-turbulence energy transfer in the vertical direction. This buoyancy effect is especially important in the self-propelled case because it allows regions of positive and negative momentum to become decoupled in the vertical direction and decay with different rates. The vertical wake thickness is found to be larger in self-propelled wakes. The role of internal waves in the energetics is determined and it is found that, later in the evolution, they can become a dominant term in the balance of turbulent kinetic energy. The non-equilibrium stage, known to exist for towed wakes, is also shown to exist for self-propelled wakes. Both the towed and self-propelled wakes, at Re = 50000, are found to exhibit a time span when, although the turbulence is strongly stratified as indicated by small Froude number, the turbulent dissipation rate decays according to inertial scaling.
Self-excited oscillations in three-dimensional collapsible tubes: simulating their onset and large-amplitude oscillations
- MATTHIAS HEIL, JONATHAN BOYLE
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- 13 April 2010, pp. 405-426
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We employ numerical simulations to explore the development of flow-induced self-excited oscillations in three-dimensional collapsible tubes which are subject to boundary conditions (flow rate prescribed at the outflow boundary) that encourage the development of high-frequency oscillations via an instability mechanism originally proposed by Jensen & Heil (J. Fluid Mech., vol. 481, 2003, p. 235). The simulations show that self-excited oscillations tend to arise preferentially from steady equilibrium configurations in which the tube is buckled non-axisymmetrically. We follow the growing oscillations into the large-amplitude regime and show that short tubes tend to approach an approximately axisymmetric equilibrium configuration in which the oscillations decay, whereas sufficiently long tubes develop sustained large-amplitude limit-cycle oscillations. The period of the oscillations and the critical Reynolds number beyond which their amplitude grows are found to be in good agreement with theoretical scaling estimates.
Rapid granular flows down inclined planar chutes. Part 1. Steady flows, multiple solutions and existence domains
- MARK J. WOODHOUSE, ANDREW J. HOGG, ALISTAIR A. SELLAR
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- 19 May 2010, pp. 427-460
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The highly agitated flow of grains down an inclined chute is modelled using a kinetic theory for inelastic collisions. Solutions corresponding to steady, fully developed flows are obtained by solving numerically a nonlinear system of ordinary differential equations using a highly accurate pseudospectral method based on mapped Chebyshev polynomials. The solutions are characterized by introducing macroscopic, depth-integrated variables representing the mass flux of flowing material per unit width, its centre-of-mass and the mass supported within the flowing layer, and the influence of the controlling parameters on these solutions is investigated. It is shown that, in certain regions of parameter space, multiple steady solutions can be found for a specified mass flux of material. An asymptotic analysis of the governing equations, appropriate to highly agitated flows, is also developed and these results aid in the demarcation of domains in parameter space where steady solutions can be obtained.
Rapid granular flows down inclined planar chutes. Part 2. Linear stability analysis of steady flow solutions
- MARK J. WOODHOUSE, ANDREW J. HOGG
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- 19 May 2010, pp. 461-488
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The linear stability of steady solutions for a rapid granular flow down an inclined chute, modelled using a kinetic theory continuum model, is analysed. The previous studies of Forterre & Pouliquen (J. Fluid Mech., vol. 467, 2002, p. 361) and Mitarai & Nakanishi (J. Fluid Mech., vol. 507, 2004, p. 309) are extended by considering fully three-dimensional perturbations, allowing variations in both the cross-slope and downslope directions, as well as normal to the base. Our results demonstrate the existence of three qualitatively different unstable perturbations, each of which can be the most rapidly growing instability for different steady flows. By considering the linear stability of many steady solutions along macroscopic flow curves, we show that linear stability occurs in only a small part of parameter space, and furthermore the regions of linear instability do not correlate with density inversion of the underlying steady solutions. Our results suggest that inelastic clustering is the dominant instability mechanism.
Effective slip in pressure-driven flow past super-hydrophobic stripes
- A. V. BELYAEV, O. I. VINOGRADOVA
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- 19 May 2010, pp. 489-499
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A super-hydrophobic array of grooves containing trapped gas (stripes) has the potential to greatly reduce drag and enhance mixing phenomena in microfluidic devices. Recent work has focused on idealized cases of stick-perfect slip stripes. Here, we analyse the experimentally more relevant situation of a pressure-driven flow past striped slip-stick surfaces with arbitrary local slip at the gas sectors. We derive approximate formulas for maximal (longitudinal) and minimal (transverse) directional effective slip lengths that are in a good agreement with the exact numerical solution for any surface slip fraction. By representing eigenvalues of the slip length tensor, we obtain the effective slip for any orientation of stripes with respect to the mean flow. Our results imply that flow past stripes is controlled by the ratio of the local slip length to texture size. In the case of a large (compared to the texture period) slip at the gas areas, surface anisotropy leads to a tensorial effective slip, by attaining the values predicted earlier for a perfect local slip. Both effective slip lengths and anisotropy of the flow decrease when local slip becomes of the order of texture period. In the case of a small slip, we predict simple surface-averaged isotropic flows (independent of orientation).
Viscous fingering of a miscible reactive A + B → C interface: a linear stability analysis
- S. H. HEJAZI, P. M. J. TREVELYAN, J. AZAIEZ, A. DE WIT
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- 15 April 2010, pp. 501-528
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When one solution of reactant A is displacing another miscible solution of reactant B, a miscible product C can be generated in the contact zone if a simple A + B → C chemical reaction takes place. Depending on the relative effect of A, B and C on the viscosity, different viscous fingering (VF) instabilities can be observed. In this context, a linear stability analysis of this reaction–diffusion–convection problem under the quasi-steady-state approximation is performed to classify the various possible instability scenarios. To do so, we determine the criteria for an instability, obtain dispersion curves both at initial contact time using an analytical implicit solution and at later times via numerical stability analysis. Along with recovering known results for non-reactive systems where the displacement of a more viscous fluid by a less viscous one leads to a VF instability, it is found that in the presence of a chemical reaction, injecting a more viscous fluid into a less viscous fluid can also lead to instabilities. This occurs when the chemical reaction leads to the build up of non-monotonic viscosity profiles. Various instability scenarios are classified in a parameter plane spanned by Rb and Rc representing the log-mobility ratios of the viscosities of the B and C solution respectively with respect to that of the injected solution of A. A parametric study of the influence on stability of the Damköhler number and of the time elapsed after contact of the two reactive solutions is also conducted.